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Mirrors > Home > MPE Home > Th. List > f1orn | Structured version Visualization version GIF version |
Description: A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
f1orn | ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dff1o2 6825 | . 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹)) | |
2 | eqid 2731 | . . 3 ⊢ ran 𝐹 = ran 𝐹 | |
3 | df-3an 1089 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹) ∧ ran 𝐹 = ran 𝐹)) | |
4 | 2, 3 | mpbiran2 708 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = ran 𝐹) ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) |
5 | 1, 4 | bitri 274 | 1 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ◡ccnv 5668 ran crn 5670 Fun wfun 6526 Fn wfn 6527 –1-1-onto→wf1o 6531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-in 3951 df-ss 3961 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 |
This theorem is referenced by: f1f1orn 6831 infdifsn 9634 efopnlem2 26094 cycpmcl 32146 |
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